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A note on families of generalized Nörlund matrices as bounded operators on lp; pp. 137–145

Full article in PDF format | doi: 10.3176/proc.2009.3.01

Ulrich Stadtmüller, Anne Tali

We deal with generalized Nörlund matrices A = (N, pn, qn) defined by means of two non-negative sequences (pn) and (qn) with p0, q0 > 0. We are interested in simple conditions such that the associated non-negative triangular matrix A = (ank) is a bounded linear operator on lp (1 < p < ¥). Using results of D. Borwein (Canad. Math. Bull., 1998, 41, 10–14), we provide sufficient conditions and bounds for the norm ||A ||p. Our main question is whether certain families of generalized Nörlund matrices Aα = (N, pαn, qn) studied by different authors (see, e.g., Anal. Math., 2003, 29, 227–242; Math. Z., 1993, 214, 273–286) are bounded linear operators on lp. These matrices need not satisfy the sufficient conditions given by Borwein in the paper mentioned above. Explicit bounds for the norms ||Aα ||p are given.

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